\(\renewcommand{\hat}[1]{\widehat{#1}}\)

Shared Qs (024)


  1. Question

    Consider the right triangle with lengths 20, 21, 29 and acute angles \(\phi\) and \(\theta\), where \(\theta<\phi\). Find all the trigonometric ratios.

    plot of chunk unnamed-chunk-2



    Solution


  2. Question

    Consider the right triangle with lengths 5, 12, 13 and acute angles \(\phi\) and \(\theta\), where \(\theta<\phi\). Find all the trigonometric ratios.

    plot of chunk unnamed-chunk-2



    Solution


  3. Question

    Consider the right triangle with lengths 12, 35, 37 and acute angles \(\phi\) and \(\theta\), where \(\theta<\phi\). Find all the trigonometric ratios.

    plot of chunk unnamed-chunk-2



    Solution


  4. Question

    Consider the right triangle with lengths 11, 60, 61 and acute angles \(\phi\) and \(\theta\), where \(\theta<\phi\). Find all the trigonometric ratios.

    plot of chunk unnamed-chunk-2



    Solution


  5. Question

    Consider the right triangle with lengths 28, 45, 53 and acute angles \(\phi\) and \(\theta\), where \(\theta<\phi\). Find all the trigonometric ratios.

    plot of chunk unnamed-chunk-2



    Solution


  6. Question

    Consider the right triangle with lengths 8, 15, 17 and acute angles \(\phi\) and \(\theta\), where \(\theta<\phi\). Find all the trigonometric ratios.

    plot of chunk unnamed-chunk-2



    Solution


  7. Question

    Consider the right triangle with lengths 65, 72, 97 and acute angles \(\phi\) and \(\theta\), where \(\theta<\phi\). Find all the trigonometric ratios.

    plot of chunk unnamed-chunk-2



    Solution


  8. Question

    Consider the right triangle with lengths 3, 4, 5 and acute angles \(\phi\) and \(\theta\), where \(\theta<\phi\). Find all the trigonometric ratios.

    plot of chunk unnamed-chunk-2



    Solution


  9. Question

    Consider the right triangle with lengths 11, 60, 61 and acute angles \(\phi\) and \(\theta\), where \(\theta<\phi\). Find all the trigonometric ratios.

    plot of chunk unnamed-chunk-2



    Solution


  10. Question

    Consider the right triangle with lengths 3, 4, 5 and acute angles \(\phi\) and \(\theta\), where \(\theta<\phi\). Find all the trigonometric ratios.

    plot of chunk unnamed-chunk-2



    Solution


  11. Question

    An adventurer, crossing a vast (flat) desert, approaches a distant monument. When the adventurer first notices the monument, the viewing angle (the angle of elevation to the top of the monument) is 0.22 radians.

    The adventurer walks toward the monument, getting 1200 meters closer; the viewing angle increases to 1.45 radians.

    How tall is the monument (in meters)?

    plot of chunk unnamed-chunk-2

    (Figure not drawn to scale.)

    PLEASE ROUND YOUR ANSWER TO 2 SIGNIFICANT FIGURES.


    Solution


  12. Question

    An adventurer, crossing a vast (flat) desert, approaches a distant monument. When the adventurer first notices the monument, the viewing angle (the angle of elevation to the top of the monument) is 0.25 radians.

    The adventurer walks toward the monument, getting 260 meters closer; the viewing angle increases to 0.87 radians.

    How tall is the monument (in meters)?

    plot of chunk unnamed-chunk-2

    (Figure not drawn to scale.)

    PLEASE ROUND YOUR ANSWER TO 2 SIGNIFICANT FIGURES.


    Solution


  13. Question

    An adventurer, crossing a vast (flat) desert, approaches a distant monument. When the adventurer first notices the monument, the viewing angle (the angle of elevation to the top of the monument) is 0.17 radians.

    The adventurer walks toward the monument, getting 520 meters closer; the viewing angle increases to 0.76 radians.

    How tall is the monument (in meters)?

    plot of chunk unnamed-chunk-2

    (Figure not drawn to scale.)

    PLEASE ROUND YOUR ANSWER TO 2 SIGNIFICANT FIGURES.


    Solution


  14. Question

    An adventurer, crossing a vast (flat) desert, approaches a distant monument. When the adventurer first notices the monument, the viewing angle (the angle of elevation to the top of the monument) is 0.3 radians.

    The adventurer walks toward the monument, getting 980 meters closer; the viewing angle increases to 0.76 radians.

    How tall is the monument (in meters)?

    plot of chunk unnamed-chunk-2

    (Figure not drawn to scale.)

    PLEASE ROUND YOUR ANSWER TO 2 SIGNIFICANT FIGURES.


    Solution


  15. Question

    An adventurer, crossing a vast (flat) desert, approaches a distant monument. When the adventurer first notices the monument, the viewing angle (the angle of elevation to the top of the monument) is 0.17 radians.

    The adventurer walks toward the monument, getting 260 meters closer; the viewing angle increases to 0.49 radians.

    How tall is the monument (in meters)?

    plot of chunk unnamed-chunk-2

    (Figure not drawn to scale.)

    PLEASE ROUND YOUR ANSWER TO 2 SIGNIFICANT FIGURES.


    Solution


  16. Question

    An adventurer, crossing a vast (flat) desert, approaches a distant monument. When the adventurer first notices the monument, the viewing angle (the angle of elevation to the top of the monument) is 0.25 radians.

    The adventurer walks toward the monument, getting 1700 meters closer; the viewing angle increases to 0.98 radians.

    How tall is the monument (in meters)?

    plot of chunk unnamed-chunk-2

    (Figure not drawn to scale.)

    PLEASE ROUND YOUR ANSWER TO 2 SIGNIFICANT FIGURES.


    Solution


  17. Question

    An adventurer, crossing a vast (flat) desert, approaches a distant monument. When the adventurer first notices the monument, the viewing angle (the angle of elevation to the top of the monument) is 0.32 radians.

    The adventurer walks toward the monument, getting 1300 meters closer; the viewing angle increases to 0.89 radians.

    How tall is the monument (in meters)?

    plot of chunk unnamed-chunk-2

    (Figure not drawn to scale.)

    PLEASE ROUND YOUR ANSWER TO 2 SIGNIFICANT FIGURES.


    Solution


  18. Question

    An adventurer, crossing a vast (flat) desert, approaches a distant monument. When the adventurer first notices the monument, the viewing angle (the angle of elevation to the top of the monument) is 0.28 radians.

    The adventurer walks toward the monument, getting 880 meters closer; the viewing angle increases to 0.96 radians.

    How tall is the monument (in meters)?

    plot of chunk unnamed-chunk-2

    (Figure not drawn to scale.)

    PLEASE ROUND YOUR ANSWER TO 2 SIGNIFICANT FIGURES.


    Solution


  19. Question

    An adventurer, crossing a vast (flat) desert, approaches a distant monument. When the adventurer first notices the monument, the viewing angle (the angle of elevation to the top of the monument) is 0.33 radians.

    The adventurer walks toward the monument, getting 1200 meters closer; the viewing angle increases to 0.72 radians.

    How tall is the monument (in meters)?

    plot of chunk unnamed-chunk-2

    (Figure not drawn to scale.)

    PLEASE ROUND YOUR ANSWER TO 2 SIGNIFICANT FIGURES.


    Solution


  20. Question

    An adventurer, crossing a vast (flat) desert, approaches a distant monument. When the adventurer first notices the monument, the viewing angle (the angle of elevation to the top of the monument) is 0.13 radians.

    The adventurer walks toward the monument, getting 1300 meters closer; the viewing angle increases to 1.43 radians.

    How tall is the monument (in meters)?

    plot of chunk unnamed-chunk-2

    (Figure not drawn to scale.)

    PLEASE ROUND YOUR ANSWER TO 2 SIGNIFICANT FIGURES.


    Solution


  21. Question

    Find \(x\). The angles are measured in radians.

    plot of chunk unnamed-chunk-2

    The tolerance is \(\pm 0.01\).


    Solution


  22. Question

    Find \(x\). The angles are measured in radians.

    plot of chunk unnamed-chunk-2

    The tolerance is \(\pm 0.01\).


    Solution


  23. Question

    Find \(x\). The angles are measured in radians.

    plot of chunk unnamed-chunk-2

    The tolerance is \(\pm 0.01\).


    Solution


  24. Question

    Find \(x\). The angles are measured in radians.

    plot of chunk unnamed-chunk-2

    The tolerance is \(\pm 0.01\).


    Solution


  25. Question

    Find \(x\). The angles are measured in radians.

    plot of chunk unnamed-chunk-2

    The tolerance is \(\pm 0.01\).


    Solution


  26. Question

    Find \(x\). The angles are measured in radians.

    plot of chunk unnamed-chunk-2

    The tolerance is \(\pm 0.01\).


    Solution


  27. Question

    Find \(x\). The angles are measured in radians.

    plot of chunk unnamed-chunk-2

    The tolerance is \(\pm 0.01\).


    Solution


  28. Question

    Find \(x\). The angles are measured in radians.

    plot of chunk unnamed-chunk-2

    The tolerance is \(\pm 0.01\).


    Solution


  29. Question

    Find \(x\). The angles are measured in radians.

    plot of chunk unnamed-chunk-2

    The tolerance is \(\pm 0.01\).


    Solution


  30. Question

    Find \(x\). The angles are measured in radians.

    plot of chunk unnamed-chunk-2

    The tolerance is \(\pm 0.01\).


    Solution


  31. Question

    A ship sailed 7.4 miles in a direction of 30° North of East. The ship turned and sailed 9.4 miles in a direction 74° North of East.

    plot of chunk unnamed-chunk-2

    The ship’s captain suffers a terrible accident. The first officer radios for help and sets anchor, waiting for help while administering first aid.

    A helicopter, starting where the ship started, wishes to fly directly to the ship. What direction (\(\theta\) rounded to nearest degree) and distance (\(r\) rounded to nearest tenth of a mile) should the helicopter fly?

    plot of chunk unnamed-chunk-3



    Solution


  32. Question

    A ship sailed 7.8 miles in a direction of 19° North of East. The ship turned and sailed 6.7 miles in a direction 41° North of East.

    plot of chunk unnamed-chunk-2

    The ship’s captain suffers a terrible accident. The first officer radios for help and sets anchor, waiting for help while administering first aid.

    A helicopter, starting where the ship started, wishes to fly directly to the ship. What direction (\(\theta\) rounded to nearest degree) and distance (\(r\) rounded to nearest tenth of a mile) should the helicopter fly?

    plot of chunk unnamed-chunk-3



    Solution


  33. Question

    A ship sailed 8.5 miles in a direction of 13° North of East. The ship turned and sailed 6.4 miles in a direction 38° North of East.

    plot of chunk unnamed-chunk-2

    The ship’s captain suffers a terrible accident. The first officer radios for help and sets anchor, waiting for help while administering first aid.

    A helicopter, starting where the ship started, wishes to fly directly to the ship. What direction (\(\theta\) rounded to nearest degree) and distance (\(r\) rounded to nearest tenth of a mile) should the helicopter fly?

    plot of chunk unnamed-chunk-3



    Solution


  34. Question

    A ship sailed 5.5 miles in a direction of 30° North of East. The ship turned and sailed 4.7 miles in a direction 71° North of East.

    plot of chunk unnamed-chunk-2

    The ship’s captain suffers a terrible accident. The first officer radios for help and sets anchor, waiting for help while administering first aid.

    A helicopter, starting where the ship started, wishes to fly directly to the ship. What direction (\(\theta\) rounded to nearest degree) and distance (\(r\) rounded to nearest tenth of a mile) should the helicopter fly?

    plot of chunk unnamed-chunk-3



    Solution


  35. Question

    A ship sailed 9.3 miles in a direction of 28° North of East. The ship turned and sailed 9.8 miles in a direction 74° North of East.

    plot of chunk unnamed-chunk-2

    The ship’s captain suffers a terrible accident. The first officer radios for help and sets anchor, waiting for help while administering first aid.

    A helicopter, starting where the ship started, wishes to fly directly to the ship. What direction (\(\theta\) rounded to nearest degree) and distance (\(r\) rounded to nearest tenth of a mile) should the helicopter fly?

    plot of chunk unnamed-chunk-3



    Solution


  36. Question

    A ship sailed 5.6 miles in a direction of 16° North of East. The ship turned and sailed 4.9 miles in a direction 63° North of East.

    plot of chunk unnamed-chunk-2

    The ship’s captain suffers a terrible accident. The first officer radios for help and sets anchor, waiting for help while administering first aid.

    A helicopter, starting where the ship started, wishes to fly directly to the ship. What direction (\(\theta\) rounded to nearest degree) and distance (\(r\) rounded to nearest tenth of a mile) should the helicopter fly?

    plot of chunk unnamed-chunk-3



    Solution


  37. Question

    A ship sailed 7.2 miles in a direction of 16° North of East. The ship turned and sailed 6.6 miles in a direction 54° North of East.

    plot of chunk unnamed-chunk-2

    The ship’s captain suffers a terrible accident. The first officer radios for help and sets anchor, waiting for help while administering first aid.

    A helicopter, starting where the ship started, wishes to fly directly to the ship. What direction (\(\theta\) rounded to nearest degree) and distance (\(r\) rounded to nearest tenth of a mile) should the helicopter fly?

    plot of chunk unnamed-chunk-3



    Solution


  38. Question

    A ship sailed 2.9 miles in a direction of 22° North of East. The ship turned and sailed 3.5 miles in a direction 73° North of East.

    plot of chunk unnamed-chunk-2

    The ship’s captain suffers a terrible accident. The first officer radios for help and sets anchor, waiting for help while administering first aid.

    A helicopter, starting where the ship started, wishes to fly directly to the ship. What direction (\(\theta\) rounded to nearest degree) and distance (\(r\) rounded to nearest tenth of a mile) should the helicopter fly?

    plot of chunk unnamed-chunk-3



    Solution


  39. Question

    A ship sailed 9.3 miles in a direction of 30° North of East. The ship turned and sailed 7.6 miles in a direction 55° North of East.

    plot of chunk unnamed-chunk-2

    The ship’s captain suffers a terrible accident. The first officer radios for help and sets anchor, waiting for help while administering first aid.

    A helicopter, starting where the ship started, wishes to fly directly to the ship. What direction (\(\theta\) rounded to nearest degree) and distance (\(r\) rounded to nearest tenth of a mile) should the helicopter fly?

    plot of chunk unnamed-chunk-3



    Solution


  40. Question

    A ship sailed 2.7 miles in a direction of 27° North of East. The ship turned and sailed 3.3 miles in a direction 53° North of East.

    plot of chunk unnamed-chunk-2

    The ship’s captain suffers a terrible accident. The first officer radios for help and sets anchor, waiting for help while administering first aid.

    A helicopter, starting where the ship started, wishes to fly directly to the ship. What direction (\(\theta\) rounded to nearest degree) and distance (\(r\) rounded to nearest tenth of a mile) should the helicopter fly?

    plot of chunk unnamed-chunk-3



    Solution


  41. Question

    A line forms an angle of inclination of \(\theta=-1.12\) radians with the horizontal axis. What is the slope of the line?

    plot of chunk unnamed-chunk-2

    In other words, the line could be represented with a linear equation, \(y=mx+b\), and your goal is to determine \(m\).

    The tolerance is \(\pm 0.01\).


    Solution


  42. Question

    A line forms an angle of inclination of \(\theta=1.01\) radians with the horizontal axis. What is the slope of the line?

    plot of chunk unnamed-chunk-2

    In other words, the line could be represented with a linear equation, \(y=mx+b\), and your goal is to determine \(m\).

    The tolerance is \(\pm 0.01\).


    Solution


  43. Question

    A line forms an angle of inclination of \(\theta=-0.61\) radians with the horizontal axis. What is the slope of the line?

    plot of chunk unnamed-chunk-2

    In other words, the line could be represented with a linear equation, \(y=mx+b\), and your goal is to determine \(m\).

    The tolerance is \(\pm 0.01\).


    Solution


  44. Question

    A line forms an angle of inclination of \(\theta=0.99\) radians with the horizontal axis. What is the slope of the line?

    plot of chunk unnamed-chunk-2

    In other words, the line could be represented with a linear equation, \(y=mx+b\), and your goal is to determine \(m\).

    The tolerance is \(\pm 0.01\).


    Solution


  45. Question

    A line forms an angle of inclination of \(\theta=-1.07\) radians with the horizontal axis. What is the slope of the line?

    plot of chunk unnamed-chunk-2

    In other words, the line could be represented with a linear equation, \(y=mx+b\), and your goal is to determine \(m\).

    The tolerance is \(\pm 0.01\).


    Solution


  46. Question

    A line forms an angle of inclination of \(\theta=-0.56\) radians with the horizontal axis. What is the slope of the line?

    plot of chunk unnamed-chunk-2

    In other words, the line could be represented with a linear equation, \(y=mx+b\), and your goal is to determine \(m\).

    The tolerance is \(\pm 0.01\).


    Solution


  47. Question

    A line forms an angle of inclination of \(\theta=1.14\) radians with the horizontal axis. What is the slope of the line?

    plot of chunk unnamed-chunk-2

    In other words, the line could be represented with a linear equation, \(y=mx+b\), and your goal is to determine \(m\).

    The tolerance is \(\pm 0.01\).


    Solution


  48. Question

    A line forms an angle of inclination of \(\theta=0.75\) radians with the horizontal axis. What is the slope of the line?

    plot of chunk unnamed-chunk-2

    In other words, the line could be represented with a linear equation, \(y=mx+b\), and your goal is to determine \(m\).

    The tolerance is \(\pm 0.01\).


    Solution


  49. Question

    A line forms an angle of inclination of \(\theta=-1.13\) radians with the horizontal axis. What is the slope of the line?

    plot of chunk unnamed-chunk-2

    In other words, the line could be represented with a linear equation, \(y=mx+b\), and your goal is to determine \(m\).

    The tolerance is \(\pm 0.01\).


    Solution


  50. Question

    A line forms an angle of inclination of \(\theta=0.83\) radians with the horizontal axis. What is the slope of the line?

    plot of chunk unnamed-chunk-2

    In other words, the line could be represented with a linear equation, \(y=mx+b\), and your goal is to determine \(m\).

    The tolerance is \(\pm 0.01\).


    Solution


  51. Question

    On steep streets, you may see signs that indicate the grade, which is usually shown as a percentage. The grade is equivalent to slope, written as a percentage.

    The highest grade of the Ashuwillticook Rail Trail is 2.5%. The Mohawk Trail (Route 2), near the hairpin turn, has sections of 8% grade. Notch Road, heading up Mount Greylock, has grades as high as 12%. Circuit Road, below Berry Pond in Pittsfield State Forest, has a section of 16% grade. Thunderbolt Ski Trail’s steepest section has 35% grade.

    plot of chunk unnamed-chunk-2

    A grade of 9.5% implies that for every 100 feet of horizontal travel (as seen on a map), the elevation changes by 9.5 feet. We wish to determine the angle of inclination (in degrees) of a 9.5% grade.

    plot of chunk unnamed-chunk-3

    Find \(\theta\), the angle of inclination in degrees. The tolerance is \(\pm0.1^\circ\).


    Solution


  52. Question

    On steep streets, you may see signs that indicate the grade, which is usually shown as a percentage. The grade is equivalent to slope, written as a percentage.

    The highest grade of the Ashuwillticook Rail Trail is 2.5%. The Mohawk Trail (Route 2), near the hairpin turn, has sections of 8% grade. Notch Road, heading up Mount Greylock, has grades as high as 12%. Circuit Road, below Berry Pond in Pittsfield State Forest, has a section of 16% grade. Thunderbolt Ski Trail’s steepest section has 35% grade.

    plot of chunk unnamed-chunk-2

    A grade of 21.3% implies that for every 100 feet of horizontal travel (as seen on a map), the elevation changes by 21.3 feet. We wish to determine the angle of inclination (in degrees) of a 21.3% grade.

    plot of chunk unnamed-chunk-3

    Find \(\theta\), the angle of inclination in degrees. The tolerance is \(\pm0.1^\circ\).


    Solution


  53. Question

    On steep streets, you may see signs that indicate the grade, which is usually shown as a percentage. The grade is equivalent to slope, written as a percentage.

    The highest grade of the Ashuwillticook Rail Trail is 2.5%. The Mohawk Trail (Route 2), near the hairpin turn, has sections of 8% grade. Notch Road, heading up Mount Greylock, has grades as high as 12%. Circuit Road, below Berry Pond in Pittsfield State Forest, has a section of 16% grade. Thunderbolt Ski Trail’s steepest section has 35% grade.

    plot of chunk unnamed-chunk-2

    A grade of 10.7% implies that for every 100 feet of horizontal travel (as seen on a map), the elevation changes by 10.7 feet. We wish to determine the angle of inclination (in degrees) of a 10.7% grade.

    plot of chunk unnamed-chunk-3

    Find \(\theta\), the angle of inclination in degrees. The tolerance is \(\pm0.1^\circ\).


    Solution


  54. Question

    On steep streets, you may see signs that indicate the grade, which is usually shown as a percentage. The grade is equivalent to slope, written as a percentage.

    The highest grade of the Ashuwillticook Rail Trail is 2.5%. The Mohawk Trail (Route 2), near the hairpin turn, has sections of 8% grade. Notch Road, heading up Mount Greylock, has grades as high as 12%. Circuit Road, below Berry Pond in Pittsfield State Forest, has a section of 16% grade. Thunderbolt Ski Trail’s steepest section has 35% grade.

    plot of chunk unnamed-chunk-2

    A grade of 26.5% implies that for every 100 feet of horizontal travel (as seen on a map), the elevation changes by 26.5 feet. We wish to determine the angle of inclination (in degrees) of a 26.5% grade.

    plot of chunk unnamed-chunk-3

    Find \(\theta\), the angle of inclination in degrees. The tolerance is \(\pm0.1^\circ\).


    Solution


  55. Question

    On steep streets, you may see signs that indicate the grade, which is usually shown as a percentage. The grade is equivalent to slope, written as a percentage.

    The highest grade of the Ashuwillticook Rail Trail is 2.5%. The Mohawk Trail (Route 2), near the hairpin turn, has sections of 8% grade. Notch Road, heading up Mount Greylock, has grades as high as 12%. Circuit Road, below Berry Pond in Pittsfield State Forest, has a section of 16% grade. Thunderbolt Ski Trail’s steepest section has 35% grade.

    plot of chunk unnamed-chunk-2

    A grade of 28.6% implies that for every 100 feet of horizontal travel (as seen on a map), the elevation changes by 28.6 feet. We wish to determine the angle of inclination (in degrees) of a 28.6% grade.

    plot of chunk unnamed-chunk-3

    Find \(\theta\), the angle of inclination in degrees. The tolerance is \(\pm0.1^\circ\).


    Solution


  56. Question

    On steep streets, you may see signs that indicate the grade, which is usually shown as a percentage. The grade is equivalent to slope, written as a percentage.

    The highest grade of the Ashuwillticook Rail Trail is 2.5%. The Mohawk Trail (Route 2), near the hairpin turn, has sections of 8% grade. Notch Road, heading up Mount Greylock, has grades as high as 12%. Circuit Road, below Berry Pond in Pittsfield State Forest, has a section of 16% grade. Thunderbolt Ski Trail’s steepest section has 35% grade.

    plot of chunk unnamed-chunk-2

    A grade of 28.9% implies that for every 100 feet of horizontal travel (as seen on a map), the elevation changes by 28.9 feet. We wish to determine the angle of inclination (in degrees) of a 28.9% grade.

    plot of chunk unnamed-chunk-3

    Find \(\theta\), the angle of inclination in degrees. The tolerance is \(\pm0.1^\circ\).


    Solution


  57. Question

    On steep streets, you may see signs that indicate the grade, which is usually shown as a percentage. The grade is equivalent to slope, written as a percentage.

    The highest grade of the Ashuwillticook Rail Trail is 2.5%. The Mohawk Trail (Route 2), near the hairpin turn, has sections of 8% grade. Notch Road, heading up Mount Greylock, has grades as high as 12%. Circuit Road, below Berry Pond in Pittsfield State Forest, has a section of 16% grade. Thunderbolt Ski Trail’s steepest section has 35% grade.

    plot of chunk unnamed-chunk-2

    A grade of 19.1% implies that for every 100 feet of horizontal travel (as seen on a map), the elevation changes by 19.1 feet. We wish to determine the angle of inclination (in degrees) of a 19.1% grade.

    plot of chunk unnamed-chunk-3

    Find \(\theta\), the angle of inclination in degrees. The tolerance is \(\pm0.1^\circ\).


    Solution


  58. Question

    On steep streets, you may see signs that indicate the grade, which is usually shown as a percentage. The grade is equivalent to slope, written as a percentage.

    The highest grade of the Ashuwillticook Rail Trail is 2.5%. The Mohawk Trail (Route 2), near the hairpin turn, has sections of 8% grade. Notch Road, heading up Mount Greylock, has grades as high as 12%. Circuit Road, below Berry Pond in Pittsfield State Forest, has a section of 16% grade. Thunderbolt Ski Trail’s steepest section has 35% grade.

    plot of chunk unnamed-chunk-2

    A grade of 7.1% implies that for every 100 feet of horizontal travel (as seen on a map), the elevation changes by 7.1 feet. We wish to determine the angle of inclination (in degrees) of a 7.1% grade.

    plot of chunk unnamed-chunk-3

    Find \(\theta\), the angle of inclination in degrees. The tolerance is \(\pm0.1^\circ\).


    Solution


  59. Question

    On steep streets, you may see signs that indicate the grade, which is usually shown as a percentage. The grade is equivalent to slope, written as a percentage.

    The highest grade of the Ashuwillticook Rail Trail is 2.5%. The Mohawk Trail (Route 2), near the hairpin turn, has sections of 8% grade. Notch Road, heading up Mount Greylock, has grades as high as 12%. Circuit Road, below Berry Pond in Pittsfield State Forest, has a section of 16% grade. Thunderbolt Ski Trail’s steepest section has 35% grade.

    plot of chunk unnamed-chunk-2

    A grade of 23.8% implies that for every 100 feet of horizontal travel (as seen on a map), the elevation changes by 23.8 feet. We wish to determine the angle of inclination (in degrees) of a 23.8% grade.

    plot of chunk unnamed-chunk-3

    Find \(\theta\), the angle of inclination in degrees. The tolerance is \(\pm0.1^\circ\).


    Solution


  60. Question

    On steep streets, you may see signs that indicate the grade, which is usually shown as a percentage. The grade is equivalent to slope, written as a percentage.

    The highest grade of the Ashuwillticook Rail Trail is 2.5%. The Mohawk Trail (Route 2), near the hairpin turn, has sections of 8% grade. Notch Road, heading up Mount Greylock, has grades as high as 12%. Circuit Road, below Berry Pond in Pittsfield State Forest, has a section of 16% grade. Thunderbolt Ski Trail’s steepest section has 35% grade.

    plot of chunk unnamed-chunk-2

    A grade of 39.8% implies that for every 100 feet of horizontal travel (as seen on a map), the elevation changes by 39.8 feet. We wish to determine the angle of inclination (in degrees) of a 39.8% grade.

    plot of chunk unnamed-chunk-3

    Find \(\theta\), the angle of inclination in degrees. The tolerance is \(\pm0.1^\circ\).


    Solution


  61. Question

    A point on a Cartesian plane is given in polar coordinates: \[r = 3.53\] \[\theta=\frac{15\pi}{13}\]

    plot of chunk unnamed-chunk-2

    Please determine the rectangular coordinates (\(x\) and \(y\)).



    Solution


  62. Question

    A point on a Cartesian plane is given in polar coordinates: \[r = 3.17\] \[\theta=\frac{6\pi}{7}\]

    plot of chunk unnamed-chunk-2

    Please determine the rectangular coordinates (\(x\) and \(y\)).



    Solution


  63. Question

    A point on a Cartesian plane is given in polar coordinates: \[r = 3.41\] \[\theta=\frac{7\pi}{9}\]

    plot of chunk unnamed-chunk-2

    Please determine the rectangular coordinates (\(x\) and \(y\)).



    Solution


  64. Question

    A point on a Cartesian plane is given in polar coordinates: \[r = 4.94\] \[\theta=\frac{12\pi}{7}\]

    plot of chunk unnamed-chunk-2

    Please determine the rectangular coordinates (\(x\) and \(y\)).



    Solution


  65. Question

    A point on a Cartesian plane is given in polar coordinates: \[r = 3.14\] \[\theta=\frac{11\pi}{9}\]

    plot of chunk unnamed-chunk-2

    Please determine the rectangular coordinates (\(x\) and \(y\)).



    Solution


  66. Question

    A point on a Cartesian plane is given in polar coordinates: \[r = 3.33\] \[\theta=\frac{8\pi}{11}\]

    plot of chunk unnamed-chunk-2

    Please determine the rectangular coordinates (\(x\) and \(y\)).



    Solution


  67. Question

    A point on a Cartesian plane is given in polar coordinates: \[r = 5.66\] \[\theta=\frac{14\pi}{11}\]

    plot of chunk unnamed-chunk-2

    Please determine the rectangular coordinates (\(x\) and \(y\)).



    Solution


  68. Question

    A point on a Cartesian plane is given in polar coordinates: \[r = 4.54\] \[\theta=\frac{15\pi}{11}\]

    plot of chunk unnamed-chunk-2

    Please determine the rectangular coordinates (\(x\) and \(y\)).



    Solution


  69. Question

    A point on a Cartesian plane is given in polar coordinates: \[r = 3.52\] \[\theta=\frac{14\pi}{11}\]

    plot of chunk unnamed-chunk-2

    Please determine the rectangular coordinates (\(x\) and \(y\)).



    Solution


  70. Question

    A point on a Cartesian plane is given in polar coordinates: \[r = 4.82\] \[\theta=\frac{9\pi}{8}\]

    plot of chunk unnamed-chunk-2

    Please determine the rectangular coordinates (\(x\) and \(y\)).



    Solution


  71. Question

    A point on a Cartesian plane is given in rectangular coordinates: \[x = 3.76\] \[y=-4.34\]

    plot of chunk unnamed-chunk-2

    Please determine the polar coordinates (\(r\) and \(\theta\)), enforcing \(0\le\theta<2\pi\).



    Solution


  72. Question

    A point on a Cartesian plane is given in rectangular coordinates: \[x = -4.24\] \[y=1.38\]

    plot of chunk unnamed-chunk-2

    Please determine the polar coordinates (\(r\) and \(\theta\)), enforcing \(0\le\theta<2\pi\).



    Solution


  73. Question

    A point on a Cartesian plane is given in rectangular coordinates: \[x = 2.63\] \[y=-2.1\]

    plot of chunk unnamed-chunk-2

    Please determine the polar coordinates (\(r\) and \(\theta\)), enforcing \(0\le\theta<2\pi\).



    Solution


  74. Question

    A point on a Cartesian plane is given in rectangular coordinates: \[x = -1.34\] \[y=4.12\]

    plot of chunk unnamed-chunk-2

    Please determine the polar coordinates (\(r\) and \(\theta\)), enforcing \(0\le\theta<2\pi\).



    Solution


  75. Question

    A point on a Cartesian plane is given in rectangular coordinates: \[x = -1.41\] \[y=4.35\]

    plot of chunk unnamed-chunk-2

    Please determine the polar coordinates (\(r\) and \(\theta\)), enforcing \(0\le\theta<2\pi\).



    Solution


  76. Question

    A point on a Cartesian plane is given in rectangular coordinates: \[x = 2.48\] \[y=-3.11\]

    plot of chunk unnamed-chunk-2

    Please determine the polar coordinates (\(r\) and \(\theta\)), enforcing \(0\le\theta<2\pi\).



    Solution


  77. Question

    A point on a Cartesian plane is given in rectangular coordinates: \[x = -1.26\] \[y=-3.33\]

    plot of chunk unnamed-chunk-2

    Please determine the polar coordinates (\(r\) and \(\theta\)), enforcing \(0\le\theta<2\pi\).



    Solution


  78. Question

    A point on a Cartesian plane is given in rectangular coordinates: \[x = 1.44\] \[y=-4.42\]

    plot of chunk unnamed-chunk-2

    Please determine the polar coordinates (\(r\) and \(\theta\)), enforcing \(0\le\theta<2\pi\).



    Solution


  79. Question

    A point on a Cartesian plane is given in rectangular coordinates: \[x = 1.45\] \[y=-4.46\]

    plot of chunk unnamed-chunk-2

    Please determine the polar coordinates (\(r\) and \(\theta\)), enforcing \(0\le\theta<2\pi\).



    Solution


  80. Question

    A point on a Cartesian plane is given in rectangular coordinates: \[x = 1.68\] \[y=-3.67\]

    plot of chunk unnamed-chunk-2

    Please determine the polar coordinates (\(r\) and \(\theta\)), enforcing \(0\le\theta<2\pi\).



    Solution


  81. Question

    A 30°-60°-90° right triangle has hypotenuse length of 8.93 meters.

    plot of chunk unnamed-chunk-2

    Find the long-leg length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  82. Question

    A 45°-45°-90° right triangle has hypotenuse length of 3.6 meters.

    plot of chunk unnamed-chunk-2

    Find the leg length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  83. Question

    A 30°-60°-90° right triangle has hypotenuse length of 9.52 meters.

    plot of chunk unnamed-chunk-2

    Find the long-leg length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  84. Question

    A 45°-45°-90° right triangle has hypotenuse length of 8.75 meters.

    plot of chunk unnamed-chunk-2

    Find the leg length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  85. Question

    A 45°-45°-90° right triangle has hypotenuse length of 3.1 meters.

    plot of chunk unnamed-chunk-2

    Find the leg length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  86. Question

    A 45°-45°-90° right triangle has hypotenuse length of 7.58 meters.

    plot of chunk unnamed-chunk-2

    Find the leg length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  87. Question

    A 30°-60°-90° right triangle has hypotenuse length of 8.47 meters.

    plot of chunk unnamed-chunk-2

    Find the short-leg length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  88. Question

    A 45°-45°-90° right triangle has leg length of 8.57 meters.

    plot of chunk unnamed-chunk-2

    Find the hypotenuse length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  89. Question

    A 45°-45°-90° right triangle has hypotenuse length of 9.93 meters.

    plot of chunk unnamed-chunk-2

    Find the leg length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  90. Question

    A 45°-45°-90° right triangle has hypotenuse length of 3.52 meters.

    plot of chunk unnamed-chunk-2

    Find the leg length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  91. Question

    A right triangle with an acute angle of \(\frac{\pi}{6}\) has hypotenuse length of 8.8 meters.

    plot of chunk unnamed-chunk-2

    Find the long-leg length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  92. Question

    A right triangle with an acute angle of \(\frac{\pi}{6}\) has long-leg length of 9.66 meters.

    plot of chunk unnamed-chunk-2

    Find the short-leg length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  93. Question

    A right triangle with an acute angle of \(\frac{\pi}{4}\) has hypotenuse length of 9.69 meters.

    plot of chunk unnamed-chunk-2

    Find the leg length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  94. Question

    A right triangle with an acute angle of \(\frac{\pi}{4}\) has leg length of 6.86 meters.

    plot of chunk unnamed-chunk-2

    Find the hypotenuse length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  95. Question

    A right triangle with an acute angle of \(\frac{\pi}{4}\) has hypotenuse length of 8.38 meters.

    plot of chunk unnamed-chunk-2

    Find the leg length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  96. Question

    A right triangle with an acute angle of \(\frac{\pi}{4}\) has leg length of 4.46 meters.

    plot of chunk unnamed-chunk-2

    Find the hypotenuse length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  97. Question

    A right triangle with an acute angle of \(\frac{\pi}{6}\) has long-leg length of 2.52 meters.

    plot of chunk unnamed-chunk-2

    Find the short-leg length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  98. Question

    A right triangle with an acute angle of \(\frac{\pi}{4}\) has hypotenuse length of 7.95 meters.

    plot of chunk unnamed-chunk-2

    Find the leg length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  99. Question

    A right triangle with an acute angle of \(\frac{\pi}{4}\) has hypotenuse length of 6.34 meters.

    plot of chunk unnamed-chunk-2

    Find the leg length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  100. Question

    A right triangle with an acute angle of \(\frac{\pi}{6}\) has long-leg length of 3.84 meters.

    plot of chunk unnamed-chunk-2

    Find the hypotenuse length in meters. The tolerance is \(\pm0.01\) meters.


    Solution


  101. Question

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\cos\left(\theta\right)\).


    1. \(-1\)
    2. \(\frac{-\sqrt{3}}{2}\)
    3. \(\frac{-\sqrt{2}}{2}\)
    4. \(\frac{-1}{2}\)
    5. \(0\)
    6. \(\frac{1}{2}\)
    7. \(\frac{\sqrt{2}}{2}\)
    8. \(\frac{\sqrt{3}}{2}\)
    9. \(1\)

    Solution


  102. Question

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\cos\left(\theta\right)\).


    1. \(-1\)
    2. \(\frac{-\sqrt{3}}{2}\)
    3. \(\frac{-\sqrt{2}}{2}\)
    4. \(\frac{-1}{2}\)
    5. \(0\)
    6. \(\frac{1}{2}\)
    7. \(\frac{\sqrt{2}}{2}\)
    8. \(\frac{\sqrt{3}}{2}\)
    9. \(1\)

    Solution


  103. Question

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\sin\left(\theta\right)\).


    1. \(-1\)
    2. \(\frac{-\sqrt{3}}{2}\)
    3. \(\frac{-\sqrt{2}}{2}\)
    4. \(\frac{-1}{2}\)
    5. \(0\)
    6. \(\frac{1}{2}\)
    7. \(\frac{\sqrt{2}}{2}\)
    8. \(\frac{\sqrt{3}}{2}\)
    9. \(1\)

    Solution


  104. Question

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\sin\left(\theta\right)\).


    1. \(-1\)
    2. \(\frac{-\sqrt{3}}{2}\)
    3. \(\frac{-\sqrt{2}}{2}\)
    4. \(\frac{-1}{2}\)
    5. \(0\)
    6. \(\frac{1}{2}\)
    7. \(\frac{\sqrt{2}}{2}\)
    8. \(\frac{\sqrt{3}}{2}\)
    9. \(1\)

    Solution


  105. Question

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\sin\left(\theta\right)\).


    1. \(-1\)
    2. \(\frac{-\sqrt{3}}{2}\)
    3. \(\frac{-\sqrt{2}}{2}\)
    4. \(\frac{-1}{2}\)
    5. \(0\)
    6. \(\frac{1}{2}\)
    7. \(\frac{\sqrt{2}}{2}\)
    8. \(\frac{\sqrt{3}}{2}\)
    9. \(1\)

    Solution


  106. Question

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\cos\left(\theta\right)\).


    1. \(-1\)
    2. \(\frac{-\sqrt{3}}{2}\)
    3. \(\frac{-\sqrt{2}}{2}\)
    4. \(\frac{-1}{2}\)
    5. \(0\)
    6. \(\frac{1}{2}\)
    7. \(\frac{\sqrt{2}}{2}\)
    8. \(\frac{\sqrt{3}}{2}\)
    9. \(1\)

    Solution


  107. Question

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\sin\left(\theta\right)\).


    1. \(-1\)
    2. \(\frac{-\sqrt{3}}{2}\)
    3. \(\frac{-\sqrt{2}}{2}\)
    4. \(\frac{-1}{2}\)
    5. \(0\)
    6. \(\frac{1}{2}\)
    7. \(\frac{\sqrt{2}}{2}\)
    8. \(\frac{\sqrt{3}}{2}\)
    9. \(1\)

    Solution


  108. Question

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\cos\left(\theta\right)\).


    1. \(-1\)
    2. \(\frac{-\sqrt{3}}{2}\)
    3. \(\frac{-\sqrt{2}}{2}\)
    4. \(\frac{-1}{2}\)
    5. \(0\)
    6. \(\frac{1}{2}\)
    7. \(\frac{\sqrt{2}}{2}\)
    8. \(\frac{\sqrt{3}}{2}\)
    9. \(1\)

    Solution


  109. Question

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\cos\left(\theta\right)\).


    1. \(-1\)
    2. \(\frac{-\sqrt{3}}{2}\)
    3. \(\frac{-\sqrt{2}}{2}\)
    4. \(\frac{-1}{2}\)
    5. \(0\)
    6. \(\frac{1}{2}\)
    7. \(\frac{\sqrt{2}}{2}\)
    8. \(\frac{\sqrt{3}}{2}\)
    9. \(1\)

    Solution


  110. Question

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\cos\left(\theta\right)\).


    1. \(-1\)
    2. \(\frac{-\sqrt{3}}{2}\)
    3. \(\frac{-\sqrt{2}}{2}\)
    4. \(\frac{-1}{2}\)
    5. \(0\)
    6. \(\frac{1}{2}\)
    7. \(\frac{\sqrt{2}}{2}\)
    8. \(\frac{\sqrt{3}}{2}\)
    9. \(1\)

    Solution


  111. Question

    To evaluate the trigonometric functions by using the unit circle, it can be helpful to know the following:

    function value
    sine \(y\)
    cosine \(x\)
    tangent \(\frac{y}{x}\)
    cosecant \(\frac{1}{y}\)
    secant \(\frac{1}{x}\)
    cotangent \(\frac{x}{y}\)

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\sec\left(\theta\right)\). You might need to rationalize a denominator and/or simplify a fraction.


    1. \(-2\)
    2. \(-\sqrt{3}\)
    3. \(-\sqrt{2}\)
    4. \(\frac{-2\sqrt{3}}{3}\)
    5. \(-1\)
    6. \(\frac{-\sqrt{3}}{2}\)
    7. \(\frac{-\sqrt{2}}{2}\)
    8. \(\frac{-\sqrt{3}}{3}\)
    9. \(\frac{-1}{2}\)
    10. \(0\)
    11. \(\frac{1}{2}\)
    12. \(\frac{\sqrt{3}}{3}\)
    13. \(\frac{\sqrt{2}}{2}\)
    14. \(\frac{\sqrt{3}}{2}\)
    15. \(1\)
    16. \(\frac{2\sqrt{3}}{3}\)
    17. \(\sqrt{2}\)
    18. \(\sqrt{3}\)
    19. \(2\)
    20. \(\pm\infty\)

    Solution


  112. Question

    To evaluate the trigonometric functions by using the unit circle, it can be helpful to know the following:

    function value
    sine \(y\)
    cosine \(x\)
    tangent \(\frac{y}{x}\)
    cosecant \(\frac{1}{y}\)
    secant \(\frac{1}{x}\)
    cotangent \(\frac{x}{y}\)

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\cot\left(\theta\right)\). You might need to rationalize a denominator and/or simplify a fraction.


    1. \(-2\)
    2. \(-\sqrt{3}\)
    3. \(-\sqrt{2}\)
    4. \(\frac{-2\sqrt{3}}{3}\)
    5. \(-1\)
    6. \(\frac{-\sqrt{3}}{2}\)
    7. \(\frac{-\sqrt{2}}{2}\)
    8. \(\frac{-\sqrt{3}}{3}\)
    9. \(\frac{-1}{2}\)
    10. \(0\)
    11. \(\frac{1}{2}\)
    12. \(\frac{\sqrt{3}}{3}\)
    13. \(\frac{\sqrt{2}}{2}\)
    14. \(\frac{\sqrt{3}}{2}\)
    15. \(1\)
    16. \(\frac{2\sqrt{3}}{3}\)
    17. \(\sqrt{2}\)
    18. \(\sqrt{3}\)
    19. \(2\)
    20. \(\pm\infty\)

    Solution


  113. Question

    To evaluate the trigonometric functions by using the unit circle, it can be helpful to know the following:

    function value
    sine \(y\)
    cosine \(x\)
    tangent \(\frac{y}{x}\)
    cosecant \(\frac{1}{y}\)
    secant \(\frac{1}{x}\)
    cotangent \(\frac{x}{y}\)

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\tan\left(\theta\right)\). You might need to rationalize a denominator and/or simplify a fraction.


    1. \(-2\)
    2. \(-\sqrt{3}\)
    3. \(-\sqrt{2}\)
    4. \(\frac{-2\sqrt{3}}{3}\)
    5. \(-1\)
    6. \(\frac{-\sqrt{3}}{2}\)
    7. \(\frac{-\sqrt{2}}{2}\)
    8. \(\frac{-\sqrt{3}}{3}\)
    9. \(\frac{-1}{2}\)
    10. \(0\)
    11. \(\frac{1}{2}\)
    12. \(\frac{\sqrt{3}}{3}\)
    13. \(\frac{\sqrt{2}}{2}\)
    14. \(\frac{\sqrt{3}}{2}\)
    15. \(1\)
    16. \(\frac{2\sqrt{3}}{3}\)
    17. \(\sqrt{2}\)
    18. \(\sqrt{3}\)
    19. \(2\)
    20. \(\pm\infty\)

    Solution


  114. Question

    To evaluate the trigonometric functions by using the unit circle, it can be helpful to know the following:

    function value
    sine \(y\)
    cosine \(x\)
    tangent \(\frac{y}{x}\)
    cosecant \(\frac{1}{y}\)
    secant \(\frac{1}{x}\)
    cotangent \(\frac{x}{y}\)

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\csc\left(\theta\right)\). You might need to rationalize a denominator and/or simplify a fraction.


    1. \(-2\)
    2. \(-\sqrt{3}\)
    3. \(-\sqrt{2}\)
    4. \(\frac{-2\sqrt{3}}{3}\)
    5. \(-1\)
    6. \(\frac{-\sqrt{3}}{2}\)
    7. \(\frac{-\sqrt{2}}{2}\)
    8. \(\frac{-\sqrt{3}}{3}\)
    9. \(\frac{-1}{2}\)
    10. \(0\)
    11. \(\frac{1}{2}\)
    12. \(\frac{\sqrt{3}}{3}\)
    13. \(\frac{\sqrt{2}}{2}\)
    14. \(\frac{\sqrt{3}}{2}\)
    15. \(1\)
    16. \(\frac{2\sqrt{3}}{3}\)
    17. \(\sqrt{2}\)
    18. \(\sqrt{3}\)
    19. \(2\)
    20. \(\pm\infty\)

    Solution


  115. Question

    To evaluate the trigonometric functions by using the unit circle, it can be helpful to know the following:

    function value
    sine \(y\)
    cosine \(x\)
    tangent \(\frac{y}{x}\)
    cosecant \(\frac{1}{y}\)
    secant \(\frac{1}{x}\)
    cotangent \(\frac{x}{y}\)

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\cot\left(\theta\right)\). You might need to rationalize a denominator and/or simplify a fraction.


    1. \(-2\)
    2. \(-\sqrt{3}\)
    3. \(-\sqrt{2}\)
    4. \(\frac{-2\sqrt{3}}{3}\)
    5. \(-1\)
    6. \(\frac{-\sqrt{3}}{2}\)
    7. \(\frac{-\sqrt{2}}{2}\)
    8. \(\frac{-\sqrt{3}}{3}\)
    9. \(\frac{-1}{2}\)
    10. \(0\)
    11. \(\frac{1}{2}\)
    12. \(\frac{\sqrt{3}}{3}\)
    13. \(\frac{\sqrt{2}}{2}\)
    14. \(\frac{\sqrt{3}}{2}\)
    15. \(1\)
    16. \(\frac{2\sqrt{3}}{3}\)
    17. \(\sqrt{2}\)
    18. \(\sqrt{3}\)
    19. \(2\)
    20. \(\pm\infty\)

    Solution


  116. Question

    To evaluate the trigonometric functions by using the unit circle, it can be helpful to know the following:

    function value
    sine \(y\)
    cosine \(x\)
    tangent \(\frac{y}{x}\)
    cosecant \(\frac{1}{y}\)
    secant \(\frac{1}{x}\)
    cotangent \(\frac{x}{y}\)

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\csc\left(\theta\right)\). You might need to rationalize a denominator and/or simplify a fraction.


    1. \(-2\)
    2. \(-\sqrt{3}\)
    3. \(-\sqrt{2}\)
    4. \(\frac{-2\sqrt{3}}{3}\)
    5. \(-1\)
    6. \(\frac{-\sqrt{3}}{2}\)
    7. \(\frac{-\sqrt{2}}{2}\)
    8. \(\frac{-\sqrt{3}}{3}\)
    9. \(\frac{-1}{2}\)
    10. \(0\)
    11. \(\frac{1}{2}\)
    12. \(\frac{\sqrt{3}}{3}\)
    13. \(\frac{\sqrt{2}}{2}\)
    14. \(\frac{\sqrt{3}}{2}\)
    15. \(1\)
    16. \(\frac{2\sqrt{3}}{3}\)
    17. \(\sqrt{2}\)
    18. \(\sqrt{3}\)
    19. \(2\)
    20. \(\pm\infty\)

    Solution


  117. Question

    To evaluate the trigonometric functions by using the unit circle, it can be helpful to know the following:

    function value
    sine \(y\)
    cosine \(x\)
    tangent \(\frac{y}{x}\)
    cosecant \(\frac{1}{y}\)
    secant \(\frac{1}{x}\)
    cotangent \(\frac{x}{y}\)

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\tan\left(\theta\right)\). You might need to rationalize a denominator and/or simplify a fraction.


    1. \(-2\)
    2. \(-\sqrt{3}\)
    3. \(-\sqrt{2}\)
    4. \(\frac{-2\sqrt{3}}{3}\)
    5. \(-1\)
    6. \(\frac{-\sqrt{3}}{2}\)
    7. \(\frac{-\sqrt{2}}{2}\)
    8. \(\frac{-\sqrt{3}}{3}\)
    9. \(\frac{-1}{2}\)
    10. \(0\)
    11. \(\frac{1}{2}\)
    12. \(\frac{\sqrt{3}}{3}\)
    13. \(\frac{\sqrt{2}}{2}\)
    14. \(\frac{\sqrt{3}}{2}\)
    15. \(1\)
    16. \(\frac{2\sqrt{3}}{3}\)
    17. \(\sqrt{2}\)
    18. \(\sqrt{3}\)
    19. \(2\)
    20. \(\pm\infty\)

    Solution


  118. Question

    To evaluate the trigonometric functions by using the unit circle, it can be helpful to know the following:

    function value
    sine \(y\)
    cosine \(x\)
    tangent \(\frac{y}{x}\)
    cosecant \(\frac{1}{y}\)
    secant \(\frac{1}{x}\)
    cotangent \(\frac{x}{y}\)

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\tan\left(\theta\right)\). You might need to rationalize a denominator and/or simplify a fraction.


    1. \(-2\)
    2. \(-\sqrt{3}\)
    3. \(-\sqrt{2}\)
    4. \(\frac{-2\sqrt{3}}{3}\)
    5. \(-1\)
    6. \(\frac{-\sqrt{3}}{2}\)
    7. \(\frac{-\sqrt{2}}{2}\)
    8. \(\frac{-\sqrt{3}}{3}\)
    9. \(\frac{-1}{2}\)
    10. \(0\)
    11. \(\frac{1}{2}\)
    12. \(\frac{\sqrt{3}}{3}\)
    13. \(\frac{\sqrt{2}}{2}\)
    14. \(\frac{\sqrt{3}}{2}\)
    15. \(1\)
    16. \(\frac{2\sqrt{3}}{3}\)
    17. \(\sqrt{2}\)
    18. \(\sqrt{3}\)
    19. \(2\)
    20. \(\pm\infty\)

    Solution


  119. Question

    To evaluate the trigonometric functions by using the unit circle, it can be helpful to know the following:

    function value
    sine \(y\)
    cosine \(x\)
    tangent \(\frac{y}{x}\)
    cosecant \(\frac{1}{y}\)
    secant \(\frac{1}{x}\)
    cotangent \(\frac{x}{y}\)

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\csc\left(\theta\right)\). You might need to rationalize a denominator and/or simplify a fraction.


    1. \(-2\)
    2. \(-\sqrt{3}\)
    3. \(-\sqrt{2}\)
    4. \(\frac{-2\sqrt{3}}{3}\)
    5. \(-1\)
    6. \(\frac{-\sqrt{3}}{2}\)
    7. \(\frac{-\sqrt{2}}{2}\)
    8. \(\frac{-\sqrt{3}}{3}\)
    9. \(\frac{-1}{2}\)
    10. \(0\)
    11. \(\frac{1}{2}\)
    12. \(\frac{\sqrt{3}}{3}\)
    13. \(\frac{\sqrt{2}}{2}\)
    14. \(\frac{\sqrt{3}}{2}\)
    15. \(1\)
    16. \(\frac{2\sqrt{3}}{3}\)
    17. \(\sqrt{2}\)
    18. \(\sqrt{3}\)
    19. \(2\)
    20. \(\pm\infty\)

    Solution


  120. Question

    To evaluate the trigonometric functions by using the unit circle, it can be helpful to know the following:

    function value
    sine \(y\)
    cosine \(x\)
    tangent \(\frac{y}{x}\)
    cosecant \(\frac{1}{y}\)
    secant \(\frac{1}{x}\)
    cotangent \(\frac{x}{y}\)

    The unit circle below has a radius of length 1. The coordinates of special angles are shown.

    plot of chunk unnamed-chunk-2

    Determine \(\sec\left(\theta\right)\). You might need to rationalize a denominator and/or simplify a fraction.


    1. \(-2\)
    2. \(-\sqrt{3}\)
    3. \(-\sqrt{2}\)
    4. \(\frac{-2\sqrt{3}}{3}\)
    5. \(-1\)
    6. \(\frac{-\sqrt{3}}{2}\)
    7. \(\frac{-\sqrt{2}}{2}\)
    8. \(\frac{-\sqrt{3}}{3}\)
    9. \(\frac{-1}{2}\)
    10. \(0\)
    11. \(\frac{1}{2}\)
    12. \(\frac{\sqrt{3}}{3}\)
    13. \(\frac{\sqrt{2}}{2}\)
    14. \(\frac{\sqrt{3}}{2}\)
    15. \(1\)
    16. \(\frac{2\sqrt{3}}{3}\)
    17. \(\sqrt{2}\)
    18. \(\sqrt{3}\)
    19. \(2\)
    20. \(\pm\infty\)

    Solution


  121. Question

    An angle, \(\theta\), is shown below in standard position as a blue spiraling arrow. A radius (of length 1) is drawn at that angle. Where the radius connects to the circumference of the unit circle has Cartesian coordinates \((-0.915, -0.403)\).

    plot of chunk unnamed-chunk-2

    Calculate the trigonometric ratios of \(\theta\).



    Solution


  122. Question

    An angle, \(\theta\), is shown below in standard position as a blue spiraling arrow. A radius (of length 1) is drawn at that angle. Where the radius connects to the circumference of the unit circle has Cartesian coordinates \((-0.311, 0.95)\).

    plot of chunk unnamed-chunk-2

    Calculate the trigonometric ratios of \(\theta\).



    Solution


  123. Question

    An angle, \(\theta\), is shown below in standard position as a blue spiraling arrow. A radius (of length 1) is drawn at that angle. Where the radius connects to the circumference of the unit circle has Cartesian coordinates \((0.699, 0.715)\).

    plot of chunk unnamed-chunk-2

    Calculate the trigonometric ratios of \(\theta\).



    Solution


  124. Question

    An angle, \(\theta\), is shown below in standard position as a blue spiraling arrow. A radius (of length 1) is drawn at that angle. Where the radius connects to the circumference of the unit circle has Cartesian coordinates \((0.421, 0.907)\).

    plot of chunk unnamed-chunk-2

    Calculate the trigonometric ratios of \(\theta\).



    Solution


  125. Question

    An angle, \(\theta\), is shown below in standard position as a blue spiraling arrow. A radius (of length 1) is drawn at that angle. Where the radius connects to the circumference of the unit circle has Cartesian coordinates \((0.896, 0.443)\).

    plot of chunk unnamed-chunk-2

    Calculate the trigonometric ratios of \(\theta\).



    Solution


  126. Question

    An angle, \(\theta\), is shown below in standard position as a blue spiraling arrow. A radius (of length 1) is drawn at that angle. Where the radius connects to the circumference of the unit circle has Cartesian coordinates \((0.379, -0.926)\).

    plot of chunk unnamed-chunk-2

    Calculate the trigonometric ratios of \(\theta\).



    Solution


  127. Question

    An angle, \(\theta\), is shown below in standard position as a blue spiraling arrow. A radius (of length 1) is drawn at that angle. Where the radius connects to the circumference of the unit circle has Cartesian coordinates \((0.835, 0.55)\).

    plot of chunk unnamed-chunk-2

    Calculate the trigonometric ratios of \(\theta\).



    Solution


  128. Question

    An angle, \(\theta\), is shown below in standard position as a blue spiraling arrow. A radius (of length 1) is drawn at that angle. Where the radius connects to the circumference of the unit circle has Cartesian coordinates \((-0.739, 0.673)\).

    plot of chunk unnamed-chunk-2

    Calculate the trigonometric ratios of \(\theta\).



    Solution


  129. Question

    An angle, \(\theta\), is shown below in standard position as a blue spiraling arrow. A radius (of length 1) is drawn at that angle. Where the radius connects to the circumference of the unit circle has Cartesian coordinates \((0.655, -0.756)\).

    plot of chunk unnamed-chunk-2

    Calculate the trigonometric ratios of \(\theta\).



    Solution


  130. Question

    An angle, \(\theta\), is shown below in standard position as a blue spiraling arrow. A radius (of length 1) is drawn at that angle. Where the radius connects to the circumference of the unit circle has Cartesian coordinates \((-0.567, -0.824)\).

    plot of chunk unnamed-chunk-2

    Calculate the trigonometric ratios of \(\theta\).



    Solution


  131. Question

    Consider the parametric equations for \(0\le t\le2\pi\): \[x = \cos(8t)+\sin(7t)\] \[y = \sin(5t)+\sin(4t)\]

    plot of chunk unnamed-chunk-2


    1. A
    2. B
    3. C
    4. D

    Solution


  132. Question

    Consider the parametric equations for \(0\le t\le2\pi\): \[x = \cos(6t)+\cos(3t)\] \[y = \cos(9t)+\cos(7t)\]

    plot of chunk unnamed-chunk-2


    1. A
    2. B
    3. C
    4. D

    Solution


  133. Question

    Consider the parametric equations for \(0\le t\le2\pi\): \[x = \sin(7t)+\sin(8t)\] \[y = \cos(4t)+\sin(2t)\]

    plot of chunk unnamed-chunk-2


    1. A
    2. B
    3. C
    4. D

    Solution


  134. Question

    Consider the parametric equations for \(0\le t\le2\pi\): \[x = \sin(4t)+\sin(9t)\] \[y = \sin(6t)+\cos(8t)\]

    plot of chunk unnamed-chunk-2


    1. A
    2. B
    3. C
    4. D

    Solution


  135. Question

    Consider the parametric equations for \(0\le t\le2\pi\): \[x = \cos(3t)+\cos(7t)\] \[y = \sin(5t)+\sin(2t)\]

    plot of chunk unnamed-chunk-2


    1. A
    2. B
    3. C
    4. D

    Solution


  136. Question

    Consider the parametric equations for \(0\le t\le2\pi\): \[x = \sin(5t)+\sin(7t)\] \[y = \sin(2t)+\cos(8t)\]

    plot of chunk unnamed-chunk-2


    1. A
    2. B
    3. C
    4. D

    Solution


  137. Question

    Consider the parametric equations for \(0\le t\le2\pi\): \[x = \cos(2t)+\cos(5t)\] \[y = \cos(9t)+\cos(7t)\]

    plot of chunk unnamed-chunk-2


    1. A
    2. B
    3. C
    4. D

    Solution


  138. Question

    Consider the parametric equations for \(0\le t\le2\pi\): \[x = \sin(3t)+\sin(5t)\] \[y = \cos(8t)+\sin(9t)\]

    plot of chunk unnamed-chunk-2


    1. A
    2. B
    3. C
    4. D

    Solution


  139. Question

    Consider the parametric equations for \(0\le t\le2\pi\): \[x = \sin(9t)+\sin(3t)\] \[y = \sin(7t)+\cos(5t)\]

    plot of chunk unnamed-chunk-2


    1. A
    2. B
    3. C
    4. D

    Solution


  140. Question

    Consider the parametric equations for \(0\le t\le2\pi\): \[x = \cos(5t)+\sin(8t)\] \[y = \cos(9t)+\sin(6t)\]

    plot of chunk unnamed-chunk-2


    1. A
    2. B
    3. C
    4. D

    Solution